# L9a45

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a45 at Knotilus! L9a45 is $9^3_{7}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-2 t(2) t(1)+2 t(2) t(3) t(1)-2 t(3) t(1)+3 t(1)+2 t(2)-3 t(2) t(3)+2 t(3)-2}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)}}$ (db) Jones polynomial $-q^5+3 q^4-4 q^3+5 q^2-6 q+7-4 q^{-1} +4 q^{-2} - q^{-3} + q^{-4}$ (db) Signature 0 (db) HOMFLY-PT polynomial $a^4 z^{-2} +a^4-2 z^2 a^2-2 a^2 z^{-2} -3 a^2+z^4+z^2+ z^{-2} +2+z^4 a^{-2} +z^2 a^{-2} -z^2 a^{-4}$ (db) Kauffman polynomial $z^5 a^{-5} -2 z^3 a^{-5} +3 z^6 a^{-4} +a^4 z^4-8 z^4 a^{-4} -3 a^4 z^2+3 z^2 a^{-4} -a^4 z^{-2} +3 a^4+3 z^7 a^{-3} +a^3 z^5-8 z^5 a^{-3} +5 z^3 a^{-3} -3 a^3 z+2 a^3 z^{-1} +z^8 a^{-2} +a^2 z^6+z^6 a^{-2} +2 a^2 z^4-5 z^4 a^{-2} -6 a^2 z^2+2 z^2 a^{-2} -2 a^2 z^{-2} +5 a^2+a z^7+4 z^7 a^{-1} +a z^5-9 z^5 a^{-1} +7 z^3 a^{-1} -3 a z+2 a z^{-1} +z^8-z^6+4 z^4-4 z^2- z^{-2} +3$ (db)

### Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).
 \ r \ j \
-4-3-2-1012345χ
11         1-1
9        2 2
7       21 -1
5      32  1
3     32   -1
1    43    1
-1   36     3
-3  11      0
-5  3       3
-711        0
-91         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session.